Variational approaches to nonlinear Schrodinger and Klein-Gordon equations

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  • March 19, 2019
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  • Mukherjee, Mayukh
    • Affiliation: College of Arts and Sciences, Department of Mathematics
Abstract
  • This thesis has two chapters. In the first chapter, we investigate traveling wave solutions of nonlinear Schrodinger and Klein-Gordon equations. In the compact case, we establish existence of traveling wave solutions via energy minimization methods and prove that at least compact isotropic manifolds have genuinely traveling waves. We establish certain sharp estimates on low dimensional spheres that improve results in [T1] and carry out the subelliptic analysis for NLKG on spheres of higher dimensions. We also extend the investigation started in [T1] on compact manifolds to complete non-compact manifolds which either have a certain radial symmetry or are weakly homogeneous, using concentration-compactness type arguments. In the second chapter of the thesis, we study ground state solutions for these equations on the hyperbolic space Hn via a study of the Weinstein functional, first defined in [W]. The main result is the fact that the supremum value of the Weinstein functional on Hn is the same as that on Rn and the related fact that the supremum value of the Weinstein functional is not attained on Hn, when maximization is done in the Sobolev space H1(Hn). Lastly, we prove that a corresponding version of the conjecture will hold for the Weinstein functional with the fractional Laplacian as well. The thesis ends with four Appendices and a table of symbols, which are mainly for expository convenience.
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  • In Copyright
Advisor
  • Marzuola, Jeremy
  • Christianson, Hans
  • Metcalfe, Jason
  • Taylor, Michael Eugene
  • Eberlein, Patrick
Degree
  • Doctor of Philosophy
Degree granting institution
  • University of North Carolina at Chapel Hill Graduate School
Graduation year
  • 2015
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  • Chapel Hill, NC
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  • There are no restrictions to this item.
Date uploaded
  • August 25, 2015

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